In concluding this section, we should like to give an example which illustrates some of the ideas of the primary decomposition theorem.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

which can be viewed as a version of the Pythagorean theorem.

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

A typical application is furnished by the Arzelà – Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.

The culmination of their investigations, the Arzelà – Ascoli theorem, was a generalization of the Bolzano – Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

* The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem.

( Tychonoff's theorem, which is equivalent to the axiom of choice )

* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:

Thus, using Euler's theorem we can say that the solution must be of the form:

When we wish to assert that is a group ( for example, when stating a theorem ), we say that " is a group under ".

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

Any of the several well-known axiomatisations will do ; we assume without proof all the basic well-known results about our formalism ( such as the normal form theorem or the soundness theorem ) that we need.

In the following, we state two equivalent forms of the theorem, and show their equivalence.

Later, we prove the theorem.

While we cannot do this by simply rearranging the quantifiers, we show that it is yet enough to prove the theorem for sentences of that form.

# Finally we prove the theorem for sentences of that form.

Next we consider a generic formula φ ( which no longer uses function or constant symbols ) and apply the prenex form theorem to find a formula ψ in normal form such that φ ≡ ψ ( ψ being in normal form means that all the quantifiers in ψ, if there are any, are found at the very beginning of ψ ).

The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim-Skolem theorem, lets us sharply reduce the complexity of the generic formula for which we need to prove the theorem:

As shown by the Lemma above, we only need to prove our theorem for formulas φ in R of degree 1. φ cannot be of degree 0, since formulas in R have no free variables and don't use constant symbols.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

* To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages or a number of other methods, such as Ogden's lemma, Parikh's theorem, or using closure properties.

Furthermore, by breaking the system up into smaller components, the complexity of individual components is reduced, opening up the possibility of using techniques such as automated theorem proving to prove the correctness of crucial software subsystems.

The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors.

The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument ( changing only that the minimal counterexample requires 6 colors ) and use Kempe chains in the degree 5 situation to prove the five color theorem.

The argument above began by giving an unavoidable set of five configurations ( a single vertex with degree 1, a single vertex with degree 2, ..., a single vertex with degree 5 ) and then proceeded to show that the first 4 are reducible ; to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

The homomorphism theorem is used to prove the isomorphism theorems.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system.

Gödel's completeness theorem says that a deductive system of first-order predicate calculus is " complete " in the sense that no additional inference rules are required to prove all the logically valid formulas.

Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.